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Determination of Mass Transfer Coefficient of Methane in Heavy OilSaturated Unconsolidated Porous Media Using Constant-Pressure Technique Zehao Yang,†,‡ Mingzhe Dong,*,‡ Houjian Gong,† and Yajun Li† †

College of Petroleum Engineering, China University of Petroleum, Qingdao 266555, China Department of Chemical and Petroleum Engineering, University of Calgary, Calgary T2N 1N4, Canada

‡

S Supporting Information *

ABSTRACT: Measuring the solubility and diffusivity of methane in porous media saturated with oil is critical for analyzing the process of solvent-based recovery of heavy oil reservoir. In this study, a modified pressure-pulse decay (PPD) method is presented to measure the mass transfer coefficients of methane in porous media saturated with heavy oil as well as the solubility and diffusivity of methane in bulk oil under the same experimental conditions as those used in a comparison. The experiments are conducted under constant boundary pressure in a PVT cell, and the pressure is controlled using an auto pump, which continuously compresses the volume of an intermediate container. The accumulated volume change instead of the pressure change is recorded with time. The nonequilibrium boundary condition (BC) model is used to analyze the experimental process. The general solution of the mathematical model is derived using the Laplace transform and the approximate analytical solution of the accumulated dissolved gas is further presented for post processing of the experimental data. The mass transfer coefficients are estimated through using a multilevel single-linkage (MLSL) method to match the approximate solution with the recorded experimental data. The estimation of the parameters shows that the mass transfer coefficients are pressure dependent and that a high boundary pressure contributes to the mass transfer of methane in heavy oil due to a reduction in viscosity or density. The interface mass transfer coefficient (kint) and interface saturated concentration (cint) in porous media saturated with oil and bulk oil are almost the same under the same experimental conditions. The sensitivity analysis shows that the increase of the effective diffusion coefficient (Deff) and kint contributes to enhancing the rate of mass transfer in the oil phase and that Henry’s law constant (H) has no effect on the equilibrium time but only affects the initial saturated concentration at the interface or the total dissolved gas. The modified PPD method is robust, efficient, and easy to use in the laboratory.

1. INTRODUCTION The exploitation of heavy oil has become increasingly important in the oil and gas industry, considering the declining reserves of conventional oil and in view of many countries, such as Canada, America, Russia, and China, which have abundant resources of heavy oil and bitumen. The effective and economical recovery of these reserves can be achieved by lowering their viscosity.1 Solvent-based processes, such as the injection of a diluent (methane or CO2) and vapor extraction, have shown great potential for enhancing the heavy oil recovery. To analyze better these processes, gas solubility and diffusivity in heavy oil-saturated porous media must be determined. Thus, accurate measurements of mass transfer parameters, such as effective diffusion coefficient (Deff), interface mass-transfer coefficient (kint) and Henry’s law constant (H), become very important. The effective diffusion coefficient can be defined as follows:2,3 © XXXX American Chemical Society

Deff =

ϕD τn

(1)

where D is the diffusion coefficient, which depicts the gas diffusivity in bulk liquid, m2/s, n is a positive constant, and τ is the tortuosity of the porous media. Equation 1 illustrates the relationship between diffusion coefficient and effective diffusion coefficient. It shows that Deff is smaller than D because the path of molecular diffusion in liquid-saturated porous media is longer than that of diffusion in bulk liquid. Because the tortuosity of porous media is difficult to obtain directly, the measurement method of Deff is usually the same as that used to determine D.3−7 Received: Revised: Accepted: Published: A

March 15, 2017 June 6, 2017 June 7, 2017 June 7, 2017 DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Table 1. Studies Using the PPD Method To Measure CH4−Oil Solubility and Diffusivity authors

T (°C)

Po (MPa)

Peq (MPa)

D (×10−9m2/s)

Zhang et al.14 Sheikha et al.23 Li et al.41 Etminan et al.30 Pacheco-Roman and Hejazi27

21 75 40 30 75

3.47 4.02 6.34 5.54 4.02

3.38 3.69 6.04 5.08 3.69

4.76 0.39 2.41 0.052 0.31

kint (10−7m/s)

1.37

H (MPa/(kg/m3)

BC method

boundary pressure

0.76 0.82

equilibrium equilibrium equilibrium nonequilibrium quasi-equilibrium

change change change change change

nonequilibrium BC model. Note that the modified BC in their study is slightly different from Civan and Rasmussen’s model because the interface is assumed to be saturated with gas at the existing pressure rather than at equilibrium pressure (Peq). This difference is similar to the difference between the equilibrium BC model and the quasi-equilibrium BC model previously mentioned. Pacheco-Roman and Hejazi29 further presented the approximate analytical solution for the modified nonequilibrium BC model using the integral method.28 Henry’s law constant (H) is used to estimate the amount of gas that can be dissolved in heavy oil.39,40 Using Henry’s law constant, the relationship between the gas pressure (Po) in the solvent phase and the instantaneous equilibrium concentration (cint) at the interface can be shown as follows:

The techniques for measuring diffusion coefficient include direct methods8,9 and indirect methods.10−17 The direct methods often involve a composition measurement of the gas/liquid mixture, which is expensive, time-consuming, and more error-prone.14,15 Thus, indirect methods are often preferred to measure diffusion coefficients. The indirect methods include the pressure-pulse decay (PPD) method,12−15 the density method,10,11,18 and the electromagnetic radiation method,16 among others. Among these indirect methods, the most popular technique to determine the diffusion coefficient is the PPD method. Riazi,15 for the first time, introduced the PPD method in PVT to determine the diffusion coefficient. The theory and experiments of this method have developed several years since then. Tharanivasan et al.19 classified the most used mathematical models of the PPD method, based on the boundary condition (BC), as the equilibrium BC model, the quasi-equilibrium BC model and the nonequilibrium BC model. The BC, here, indicates the interface between the solvent gas and the heavy oil. The equilibrium BC considers that the interface is saturated with solvent gas at all times and assumes that the solvent gas pressure is maintained at equilibrium pressure14,20−22 (Peq). The equilibrium BC model is simple and easy to use. However, to meet its boundary condition, the pressure decay in the solvent gas phase is limited to a very small range; otherwise, a large error may occur when the model is used to analyze the PPD experiment. The quasiequilibrium BC model15,23−27 can address this issue because it assumes that the interface is saturated with the solvent gas at the existing pressure, instead of at the equilibrium pressure during the measurement. Sheikha et al.1 and Upreti and Mehrotra24,25 presented a numerical method for the analysis of the quasi-equilibrium BC model. Pacheco-Roman and Hejazi27 acquired the approximate analytical solution using the integral method,28 and Dong et al.26 proposed a general analytical solution for this model using the Laplace transform. Compared with the former two BC models, the study of the nonequilibrium BC model,13,19,29,30 which considers the resistance existing at the interface, is not too much. The nonequilibrium BC model was first presented by Civan and Rasmussen13 and was applied to the heavy oil−solvent interface. They used the interface mass transfer coefficient (kint) to describe the hindrance resulting from the interface. This concept is identical to the heat transfer coefficient31−33 in heat transfer engineering, which is often used to depict heat convection among different phase interfaces, such as a gas−solid interface or a gas−liquid interface. Civan and Rasmussen34 further presented the ShortTime solution and Long-Time solution to match with the early time experimental results and late-time experimental results, respectively. The reason for using two approximate analytical solutions instead of one general analytical solution to model the entire experimental PPD process is due to their applicability range.35,36 Etminan et al.37 combined nonlinear regression with the Laplace numerical inverse transform38 to deal with

cg,int =

Po H

(2)

Table 1 lists several results of studies using the PPD method to measure mass transfer coefficients of methane in bulk heavy oil or bitumen.14,23,27,30,41 In this study, we will present the general analytical solution for the nonequilibrium BC model to match with the entire PPD experimental process of solubility and diffusivity of methane in heavy oil-saturated unconsolidated porous media. The constant boundary pressure technique is used in the experiment to keep the solvent gas pressure constant and the pressure is always equal to the initial pressure or the equilibrium pressure because the mass transfer parameters, such as Deff and kint, are pressure dependent.42,43 Thus, a measurement error may always exist in the traditional PPD method, especially when the pressure decay is large in the solvent gas phase. The constant boundary pressure technique is conducted by continuously reducing the volume of the solvent gas phase when the gas dissolves into the heavy oil. The accumulated volume change data are recorded with time. The multilevel single-linkage (MLSL) method,44,45 which is a global optimization method, is used to accurately match the general solution of the nonequilibrium BC model with the experimental results to simultaneously estimate the effective diffusion coefficient (Deff), the interface mass transfer coefficient (kint), and Henry’s law constant (H). The experiments and corresponding parameter estimations for measuring the solubility and diffusivity of methane in bulk heavy oil are also presented to compare with the experimental results and estimated results in porous media. The analysis method and the constant pressure-technique are not only accurate but also robust and easy to use.

2. EXPERIMENTAL STUDY A schematic diagram of the heart part of the constant-pressure technique apparatus is shown in Figure 1. Unconsolidated sand was used as the porous media in our experiments, and it is composed of three different sizes: 100, 300, and 1050 μm. These different sand sizes were mixed together and saturated B

DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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(b) The experimental setup was then pressurized with helium gas for leak test at approximate 10 MPa. The pressure change in 3 days was less than 5 KPa. Then, the entire experimental system was vacuumed for 1 day. (c) The valve between intermediate container and cell was closed, and the solvent gas was injected into the intermediate container. The valve connecting intermediate container and the cell was opened, and the gas pressure in the cell was controlled at the desired value (Po) using an auto pump. (d) The solvent gas pressure in the cell quickly reached the desired value, then started the recording of the accumulated volume change of the auto pump. (e) After the measurement was obtained, the solvent gas was carefully released from the diffusion cell and the dissolved gas in the heavy oil was vacuumed. Then gas was injected into the intermediate container at the next pressure level, the next test began to be prepared. (f) The experimental procedure to measure the solubility and diffusivity of methane in bulk oil was also conducted by repeating the experimental steps above. The height of the bulk oil column was the same as that of the sand column. The purpose of this experiment was to compare the experimental results of diffusion processes between in oil-saturated sand and in bulk oil and to determine their mass transfer coefficients using constant-pressure technique. 2.2. Material and Pressure Control. The heavy oil used in our experiments is shown in Figure 2a. The relationship between the temperature and the viscosity of the dead oil sample was measured using a HAKKE Mars III rheometer and is shown in Figure 2b. The viscosity of the oil under our experimental temperature (40 °C) was approximately 5824 mPa·s and the density was 968.7 kg/m3. The compositions of the heavy oil are determined through distillation method and shown below (Table 2): As discussed in section 2.1, the volatile compositions of the heavy oil (gas part) will be eliminated because the system is vacuumed at the beginning of the test under the experimental temperature. The modified PPD experiments were conducted under four constant boundary gas pressures: 4, 6, 8, and 10 MPa. These values are not only the initial interface pressures but also the equilibrium pressures. The recorded volume change under

Figure 1. Schematic diagram of the heart part of the modified PPD experimental setup.

with heavy oil. The heavy oil was obtained from Henan Province, China, and its properties will be given in section 2.2 of the paper. The length of the sand column was 5.16 cm, and its diameter was 2.84 cm. The porosity for the sand column is about 0.32. A high-resolution pressure transducer was used to record the pressure in the solvent gas phase (methane) to ensure that the upper boundary pressure of the sand column was constant. The transducer can measure absolute pressure up to 16 MPa, with 0.01% accuracy and 0.0001% resolution. An intermediate container was used to store gas and butter the pressure in the sample cell to maintain a constant in the PVT cell. The pressure control portion of the apparatus includes an auto pump (Teledyne ISCO) and a high-resolution pressure transducer. The experimental setup was placed in a water bath to maintain the temperature at 40 ± 0.1 °C. A computer was used to record the cumulative volume change when the gas diffuses into the heavy oil. The test gas used in the experiment was methane with 99.99% purity. 2.1. Experimental Procedure. The experimental procedure was as follows: (a) The sand was first placed into the cell and the experimental setup was then placed in the water bath to control the temperature at 40 °C. Then heavy oil, which had been warmed to the experimental temperature, was injected into the cell from the bottom slowly. And the sand column was saturated with heavy oil in the end.

Figure 2. (a) Heavy oil obtained from Henan oil field in China. (b) Relationship between temperature and viscosity of the oil sample. C

DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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measuring and calculating every parameter on the right side of eq 4, the error during step c is controlled to less than 2%. The other error occurs at step d of the procedure, which is from the operation of maintaining a constant boundary pressure at the interface between the solvent gas and the heavy oil. The auto pump is used to inject gas into the cell to compensate for the gas dissolving into the heavy oil. A delay in the response of pressure change in cell to the gas injection from auto pump may exist. Thus, the injection rate should be controlled to a small value. The error in the experiments resulting from this delay is less than 3%.

Table 2. Compositional Analysis compositional analysis component

mass fraction

N2 H2S CO2 C1 C2 C3 C4 C5 C6 C7+

0.003 0.011 0.002 0.252 0.051 0.013 0.026 0.033 0.021 0.588

3. MATHEMATICAL MODEL 3.1. Assumptions. In the mathematical analysis of the modified PPD experiment, the following proper assumptions are used: (1) Oil is stationary, and the swelling of the oil phase due to the dissolution of methane is negligible. (2) The evaporation of the heavy oil during the experiment is negligible. (3) The concentration of the solvent gas at the interface between the heavy oil and the gas phase is equal to the saturation concentration under a given gas pressure. (4) No density-induced convection occurs during the diffusion process of methane in the heavy oil. For the PPD method, it is a common and proper practice to use the first three assumptions to simplify the model and to determine the mass transfer coefficient during the process of gas dissolution and diffusion of gas in heavy oil.14,22,27,30,47−49 The fourth assumption is used because in contrast to CO2, CH4 dissolution and diffusion do not cause an increase in the oil phase density.2,30,50,51 3.2. General Solution for the Nonequilibrium BC Model. Figure 3 depicts a schematic diagram of the

different boundary pressures can be transformed into the amount of gas dissolving into the heavy oil under standard conditions using the equation of state for real gases: ⎧ PoΔV = zΔnRT PgasΔVTst ΔVst ⎨ ⇒ ΔVst = ⇒ Δn = zPstT Vs ⎩ PstΔVst = ΔnRTst ⎪

⎪

(3)

where Po is the initial boundary pressure of the solvent phase, MPa; Pst is the pressure under standard condition, 0.1 MPa; ΔV is the accumulated volume change under constant boundary gas pressure, mL; ΔVst is the accumulated volume change under the standard condition, mL; T is the experimental temperature, K; Tst is the temperature under the standard condition, 273.15 K; Δn is the amount of accumulated solvent gas that dissolved into the heavy oil, mmol; Vs is the molar volume under the standard condition, 22.4 L/mol; and z is the gas compressibility factor of methane, which can be obtained by fitting the Carnahan−Starling equation of state to the Standing and Katz compressibility factor correlation using a simple Matlab program.46 2.3. Error Analysis. The error in the experiments may arise from two aspects in the procedure of the modified PPD experiment: The first error occurs during the step c of the procedure. When the valve connecting the intermediate container to the cell is opened, the test gas in the intermediate container immediately flows into the cell. The initial pressure of the solvent gas in the cell may not reach the desired pressure. However, after the solvent gas pressure is increased to the desired value using the auto pump, some of the gas may have dissolved into the heavy oil. The experimental error may increase when the difference between the actual initial pressure and the desired pressure of the solvent gas in the cell increases. To minimize this error, a good way is to pressurize the intermediate container to an appropriate value during the step c. This appropriate pressure can be determined through real gas state equation and the mass conservation law: Po =

zoPconVcon (Vcon + Vc)zcon

Figure 3. Schematic diagram of PVT cell and interface concentrations.

experimental cell with a constant boundary pressure and the interface resistance layer between the heavy oil and the solvent gas phase is exaggerated. The concentration (cg,int) of methane just above the interface can be obtained using Henry’s law (eq 2). In our experiment, because the constant-pressure technique is used, cg,int keeps constant. The concentration (cg) just below the interface at the beginning of the experiment is zero. As the experiment begins, the dissolved gas diffuses from the interface (z = 0) toward the bottom of the cell (z = h).

(4)

where zo is the compressibility factor of methane under the desired boundary pressure (Po); zcon is the compressibility factor of methane under the initial pressure (Pcon) in the intermediate container at step c of the procedure; Vcon is the bulk volume of the intermediate container at step c, mL; Vc is the bulk volume of the cell, mL; and Pcon is the initial pressure in the intermediate container at step c, MPa. By carefully D

DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research Fick’s second law52 describes the gas concentration distribution of one-dimensional diffusion problem as a function of spatial position and time, which can be applied to the diffusion problem in porous media saturated with heavy oil as follows: Deff

∂ 2cg ∂z

2

=

Po Ahϕ (12) H where Po is the constant boundary pressure in the cell, MPa. Because eq 11 will be further used for the post processing of the experimental results, the accuracy of this analytical result of accumulated dissolved gas (eq 11) should be verified. The nonequilibrium BC mathematical model is used again via a numerical method, namely, the finite difference method (FDM). Figure 4 describes the comparison between the M∞ = cg,intAhϕ =

∂cg (5)

∂t

where cg is the concentration beneath the interface (z = 0), mmol/mL; z is the vertical distance from the analyzed point at the interface layer, m; t is the experimental time, s; and here, Deff is the effective diffusion coefficient, which describes the diffusion process in the porous media, m2/s. The relationship between Deff and diffusion coefficient (D) is given in eq 1. When the diffusivity of methane in bulk oil is measured, the effective diffusion coefficient in eq 5 is replaced by diffusion coefficient D. The initial condition for the dissolved gas distribution in the porous media saturated with heavy oil is as follows: cg|0 ≤ z ≤ h , t = 0

(6)

Because the bottom of the cell is closed, a no flow boundary condition can be assigned to z = h: ∂cg ∂z

|z = h , t ≥ 0 = 0

Figure 4. Comparison between analytical solution and numerical solution of accumulated dissolved gas.

(7)

analytical solution and the numerical solution of the accumulated dissolved gas by assuming: the amount of total dissolved gas M∞ = 30 mmol, the effective diffusion coefficient Deff = 5.9 × 10−9 m2/s, the interface mass transfer coefficient kint = 2.546 × 10−7 m/s and the length of the sand column h = 5.16 cm. Many comparisons with other parameter values are also tested, and the results also showed an excellent agreement between the numerical solutions and the analytical solutions. It may be difficult to use the analytical solution (eq 11) of accumulated dissolved gas in practice because it takes the form of an infinite series. It is better to take out limited number of terms to approximate the analytical solution. Using the same assumption for the parameters above, it can be calculated from eq 11 that the convergence is very fast and can be extremely well approximated by the first 3 terms of the infinite series:

For the boundary condition at z = 0, the Robin boundary condition is used to describe the mass transfer process of methane from the upper side of the interface resistance layer to the area beneath of the layer, and expressed as follows: −Deff

∂cg ∂z

|z = 0, t ≥ 0 = k int(cgint − cg|z = 0, t ≥ 0 )

(8)

where kint is the interface mass transfer coefficient, m/s, and 1/ kint is the resistance of the interface.34,48 When the resistance approaches zero (kint → ∞), the nonequilibrium BC model can be transformed into the equilibrium or quasi-equilibrium BC model.47 The detailed derivation of the analytical solution (eq 9) of nonequilibrium BC model (eqs 5−8) in this paper is presented in Appendix A. The result is shown here: ⎛ cg = cg,int⎜⎜1 − ⎝

∞

∑ n=1

where ξ is equal to

2 2 2ξ cos(bn(h − z)/h)e−bn Deff t / h ⎞⎟ ⎟ (bn2 + ξ 2 + ξ)cos(bn) ⎠

hk int Deff

⎛ M t ≈ ⎜⎜1 − ⎝

(9)

and bn is the nth positive nonzero root (10)

To acquire the accumulated dissolved gas at an arbitrary time, the concentration distribution (cg) of the methane in the heavy oil is integrated from the interface to the bottom of the cell: Mt =

∫0

h

⎛ cgAϕdz = ⎜⎜1 − ⎝

∞

∑ n=1

∑ n=1

2 2 2ξ 2e−bn Deff t / h ⎞⎟ ⎟M∞ bn2(bn2 + ξ 2 + ξ) ⎠

(13)

4. RESULTS AND DISCUSSION 4.1. Parameter Estimation. The purpose of measuring the solubility and diffusivity of methane in oil-saturated porous media is to obtain the mass transfer parameters, such as the effective diffusion coefficient (Deff), the interface mass transfer coefficient (kint), and Henry’s law constant (H). The modified PPD experimental results can be obtained by recording the volume change of the auto pump, which was described in section 2. The approximate analytical solution of accumulated dissolved gas (eq 13), which describes the experimental process, was proposed in section 3. Thus, the evaluation of the parameters evaluation can be implemented by fitting the approximate analytical solution with the experimental results. A common strategy is to transform the least-squares curvefitting problem into an optimization problem. The objective function for this problem is as follows:

of: bn tan(bn) = ξ

3

2 2 2ξ 2e−bn Deff t / h ⎞⎟ ⎟M∞ bn2(bn2 + ξ 2 + ξ) ⎠

(11)

where ϕ is the porosity of the porous media; A is the area of the sand column, cm2; and M∞ is the amount of total dissolved gas when the experiment reaches equilibrium. And it is equal to E

DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 5. Measured and calculated accumulated dissolved gas curves for four experiments conducted in both oil sand and bulk oil at 40 °C. (a) Experiment 1 and 5: Po = 4.00 MPa; (b) Experiment 2 and 6: Po = 6.00 MPa.

Table 3. Results of Estimated Mass Transfer Coefficients in Bulk Oil and in Oil Saturated Sand under Different Boundary Pressures Using MLSL Method (Temperature: 40 °C) number

boundary pressure (MPa)

1 2 3 4 number

4.00 6.00 8.00 10.00 boundary pressure (MPa)

5 6 7 8

4.00 6.00 8.00 10.00

min obj(p ⃗ ) =

1 2

D (m2/s)

material bulk oil

2.38 3.47 4.82 5.73

material sand saturated with heavy oil

× × × ×

kint (m/s)

10−9 10−9 10−9 10−9 Deff (m2/s) 1.24 1.92 2.56 3.12

× × × ×

10−9 10−9 10−9 10−9

6.28 7.91 9.05 1.01

M∞ (mmol)

× 10−7 × 10−7 × 10−7 × 10−6 kint (m/s) 6.25 7.86 9.12 1.03

× × × ×

10−7 10−7 10−7 10−6

14.78 21.28 29.58 35.01 M∞ (mmol) 3.89 5.46 7.37 9.06

that of the local optimization method, which can find the minimum with adequate precision. The local optimization method we used in our study is the LM method, which is the most widely used algorithms for solving generic curve-fitting problems. Because a detailed procedure for the MLSL method has been presented in a previous paper,45 which was used to handle the dynamic parameters evaluation of shale gas transport and storage, here, we directly show the fitting results using the MLSL method in Figure 5. It can be seen that the calculated accumulated dissolved gas (Mt,comp) matches with the measured accumulated dissolved gas quite well. Figure 5 presents four groups of experimental and fitting results under 4.00 and 6.00 MPa. And the other four groups of results under 8.00 and 10.00 MPa are given in the supplemental file because the trend of these curves are similar. As shown in Figure 5, our experiments involve measuring the solubility and diffusivity of methane both in oil-saturated sand and in bulk oil. The experiments were conducted under four constant boundary pressures. The column height of oilsaturated sand and bulk oil is the same for these eight groups of experiments. It can be seen that under the same boundary conditions, the amount of dissolved gas in bulk oil is much higher than that in oil-saturated sand. This finding is because the methane is primarily dissolved in the heavy oil and the amount of adsorbed gas on the sand is negligible. It can also be found that the amount of accumulated dissolved gas increases as the boundary pressure become higher. The estimated parameters using the MLSL method for our experiments are listed in Table 3. The results reveal that the effective diffusion coefficient (Deff) increases when the boundary pressure of the sand saturated with heavy oil increases, which also applies to the measurement of diffusion coefficient (D) in bulk oil. It can be seen from Table 3 that the measured diffusion coefficient (D) and

j

∑ (Mt ,comp(p⃗ , tn) − Mt ,exp(p⃗ , tn))2 n=1

(14)

where obj is the objective function; p⃗ is the vectors of the estimated parameters and p⃗ = [Deff, kint, M∞]. Note that we do not estimate Henry’s law constant (H) directly because it can be obtained from eq 12 after the amount of total dissolved gas (M∞) is acquired through parameter estimation. The other method of obtaining M∞ is to wait for the final equilibrium of the experiment, but it usually takes a very long time. Theoretically, the time is infinite; tn is the time corresponding to the nth experimental point; Mt,comp is the computed value of the amount of accumulated dissolve gas based on the approximate analytical solution (eq 13), mmol; and Mt,exp is the experimental results for accumulated dissolve gas, mmol. Although, the recorded experimental results are accumulated volume change with time, they can be immediately transformed into the amount of substance using eq 3; j is the number of the experimental points. This parameters evaluation problem involves three unknown parameters and the objective function is complex, resulting in a nonconvex optimization problem, which means that the objective function has many local minima. Therefore, it is difficult to determine the proper starting points for the parameters evaluation if only a local optimization method, such as the Levenberg−Marquardt (LM) method,55,56 is used to find the global minimum of the objective function. In our paper, a global optimization method, called the multilevel single-linkage (MLSL) method,44,45 is used to find the global minimum of eq 14. The MLSL method combines the advantage of the stochastic method, which effectively keeps the iteration of the algorithm from reaching the local minimum point, and F

DOI: 10.1021/acs.iecr.7b01088 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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law constant and it is equal to 0.57 MPa/(kg/m3) in our experiments. To validate the proposed method in this paper, the experimental data obtained by Reamer and Sage59,60 are used to estimate diffusion coefficient (D) and interface mass transfer coefficient (kint). The experimental data by Reamer and Sage are provided in the supplemental file. In their experiments, they studied the diffusion processes of methane in propane59 and in cyclohexane.60 They determined the diffusion coefficients through direct method.26 The fitting of Reamer and Sage’s experimental results with the proposed model is shown in Figure 7. It can be seen that the calculated accumulated dissolved gas (Mt) curves match their experimental data well. The estimated results using the methods of current work and Reamer and Sage’s direct method59,60 are shown in Table 4. It can be seen that the estimated results for the diffusion coefficients for these two methods are close. In the method of current study, the constant boundary condition and nonequilibrium model are combined to estimate diffusion coefficient. It is easier to use the proposed method in this paper than to use the direct method. The direct method involves composition measurements of the gas/liquid mixtures, which is expensive, and time-consuming.14,15 4.2. Sensitivity Analysis. The parameter estimation above is an inverse problem:61,62 the mass transfer parameters were determined through using the analytical solution of the mathematical model to match with the experimental results. In this section, the sensitivity analysis is a forward problem, and it shows the effect of mass transfer parameters on the process of solubility and diffusivity of methane in heavy oil. The analysis is based on the analytical solution of the nonequilibrium BC model for accumulated dissolved gas (eq 11). The assumed model parameters for sensitivity analysis are listed in Table 5 in detail: Figure 8a describes the effect of the effective diffusion coefficient (Deff) on the process of dissolution and diffusion of methane in porous media saturated with heavy oil. The value of Deffa in the legend is given in Table 5. It can be found that larger Deff results in easier mass transfer of solvent gas within the oil phase. Figure 8b shows that larger kint leads to smaller resistance of the interface between solvent gas phase and oil phase, which means solvent gas can more easily break through the interface and dissolves into the oil phase. It can also be seen that when assumed interface mass transfer coefficient (kinta) increases by 3 times and 9 times, the corresponding two curves are almost overlapped. It displays that when kint is large, the nonequilibrium BC model can be simplified into quasiequilibrium BC model or equilibrium BC model. From Figure 8a,b, it can be found that both the rise of Deff and kint contributes to reducing the equilibrium time. A comparison of effect of Deff and kint on the whole dissolution and diffusion process is shown in Figure 8c. It delineates that the effect of kint is larger than that of Deff when the resistance of the interface between gas phase and liquid phase is large. However, when the resistance of the interface is small, the effect of Deff become more important. Figure 8d shows the effect of Henry constant on the dissolution and diffusion process. It illustrates that the amount of total dissolved gas decreases as the Henry’s law constant (H) increases. However, H has no effect on the length of the equilibrium time. From Henry’s law (eq 2), it can be found that it only has effect on the initial saturated concentration at the interface.

effective diffusion coefficient (Deff) increase as the boundary pressure increases. The reason for this phenomenon is that the dissolution of methane in the heavy oil leads to viscosity or density reduction of the liquid phase.42,43 It means the mass transfer of methane in the heavy oil is easier. And, it illustrates that D and Deff are boundary pressure dependent, explaining why a constant pressure boundary was used in this work to reduce the measurement error. As the boundary pressure increases, the interface mass transfer coefficient (kint) also increases. The relationship between boundary pressure and Deff, D, and kint reflects the same trend: higher pressure boundary is beneficial to the mass transfer of methane in heavy oil. This characteristic for the solubility and diffusivity of methane is due to the viscosity or density reduction of the heavy oil as more solvent gas dissolves into the oil phase. A comparison of the estimated result of the kint in bulk oil and porous media saturated with oil reveals that sand has almost no effect on the interface mass transfer coefficient. However, it should be noted that if the porous media is replaced by shale or coal, then the results may be different due to the adsorption effect of the organic matter. The saturated concentration on the interface can be calculated through:

c int =

M∞ Voil

⎧ M∞MCH 4 ⎪ (bulk oil) ⎪ Ah =⎨ ⎪ M∞MCH4 ⎪ Ahϕ (porous media) ⎩

(15)

2

where A is the inner area of the cell, cm ; h is the length of the column of bulk oil or porous media, cm; ϕ is the porosity of the unconsolidated porous media (sand); Voil is the volume of the heavy oil, mL; and MCH4 is the molar mass of methane, 16 g/ mol. The relationship between the saturated concentration and the boundary pressure can be obtained by substituting the amount of accumulated dissolved methane (M∞) in Table 3 to eq 15, and the results are shown in Figure 6. It shows that the

Figure 6. Estimation of Henry constant through solubility and boundary pressure measurement.

saturated concentration (cint) of methane in bulk oil and oil sand is almost the same under the same pressure. This finding also illustrates that the sand has only a very slight influence on the cint. However, if the porous media is replaced by shale or coal, the results may be different due to the adsorption effect of the organic matter. The Henry’s law constant can be acquired by linear regression57,58 of the eight points (four points for bulk oil and four points for oil sand). Using Henry’s law (eq 2), it is revealed that the slope of the fitting line in Figure 6 is Henry’s G

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Figure 7. History match of diffusion process of binary mixtures using the method of this paper (a) methane in propane;59 (b) methane in cyclohexane.60

using real gas state equation. The other one is from the delay for the response of auto pump when the pressure changes in the cell. The error is controlled less than 3% by giving a low injection rate. The mathematical model of the experiments, which considers the interface resistance, is built. The general solution of mathematical model is derived through Laplace transform and the analytical solution of the accumulated dissolved gas is further presented for post processing of experimental data. And the approximate solution of the accumulated dissolved gas is also given for easier use. A global optimization method, called multilevel single-linkage (MLSL) method, is used to estimate accurately the mass transfer parameters by matching the approximate solution with the recorded experimental data. The diffusion coefficient (D), the effective diffusion coefficient (Deff), the interface mass transfer coefficient (kint), and Henry’s law constant (H) of methane in bulk oil and oil-saturated sand are obtained via parameter estimation. It is found that diffusion coefficient (D), effective diffusion coefficient (Deff), and the interface mass transfer coefficient (kint) are pressure dependent, which means that the constant pressure technique contributes to enhancing the accuracy of the measurement. And, the higher boundary pressure will result in these parameters become larger, which means the mass transfer of methane in oil is easier. This characteristic for solubility and diffusivity of methane is due to the viscosity or density reduction of the heavy oil as more solvent gas dissolves into the oil phase. The influence of sand on the interface mass transfer coefficient (kint) and interface saturated concentration (cint) is slight. However, it should be noted that if the porous media is replaced by shale or coal, then the results may be different due to the adsorption effect of the organic matter. Through sensitivity analysis of these mass transfer coefficients based on the nonequilibrium BC model, several results can be revealed: (1) The increase of Deff and kint contributes to enhancing the rate of mass transfer in oil phase and reducing the equilibrium time of the modified PPD experiment; (2) The sensitivity of kint to the process of dissolution and diffusion of methane in oil phase declines fast as kint becomes large. When kint is not sensitive, the effect of change of kint on the process is negligible. Under this situation, the quasi-equilibrium BC model or equilibrium BC model can be used to analyze the process of solubility and diffusivity of methane in oil phase. (3) When kint is a small value, the effect of kint on the process is dominant. And, as the kint become larger, which means the resistance of the interface become smaller, the

Table 4. Comparison of the Estimated Mass Transfer Coefficients through Current Method and Direct Method (Temperature, 4.4 °C; Pressure, 2.89 MPa) experimental data of methane diffusion in propane59

experimental data of methane diffusion in cyclohexane60

parameter

current work

Reamer and Sage59

current work

Reamer and Sage60

D (m2/s) kint (m/s)

2.28 × 10−8 3.29 × 10−6

2.14 × 10−8 NA

8.99 × 10−9 2.87 × 10−6

8.72 × 10−9 NA

Table 5. Assumed Physical Parameters for Sensitivity Analysis symbol

assumed value

description

ha ϕa Aa Deffa kinta M∞a Ha Poa Peqa Ta

5.16 0.26 2.84 1.24 × 10−9 6.25 × 10−7 3.89 0.57 4 4 40

sand column length, cm porosity area of the column, cm2 effective diffusion coefficient, m2 interface mass transfer coefficient, m/s amount of total dissolved gas, mmol Henry constant, MPa/(kg/m3) initial boundary pressure, MPa equilibrium boundary pressure, MPa experimental temperature, °C

In fact, many other values for these physical parameters in Table 5 have also been tested for sensitivity, and the results reveal the same trends as depicted in Figure 8.

5. CONCLUSIONS A modified PPD method, which is based on the constantpressure experimental technique and the nonequilibrium boundary condition (BC) model, is presented to measure the solubility and diffusivity of methane in oil-saturated unconsolidated porous media. The method is also used to measure the solubility and diffusivity of methane in bulk oil under the same experimental conditions as a comparison. The modified PPD experiments were conducted under constant boundary pressure in the PVT cell. The pressure is controlled using an auto pump and the accumulated volume change instead of the pressure change was recorded with time. The experimental errors primarily arise from two steps during the experiment. One is that the initial pressure of the solvent gas in cell may not reach the desired pressure when the valve connecting the PVT cell and intermediate container was opened. This error is controlled to less than 2% by accurate measurement of dead volume and by PVT analysis of methane H

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Figure 8. Sensitivity analysis of effect of mass transfer parameters on the process of dissolution and diffusion of methane in oil phase: (a) effect of effective diffusion coefficient; (b) effect of interface mass transfer coefficient; (c) effect comparison between effective diffusion coefficient and interface mass transfer coefficient; (d) effect of henry constant.

effect of Deff becomes more important. (4) Henry’s law constant (H) has no effect on the length of the equilibrium time, and it only has effect on the initial saturated concentration at the interface or the total dissolved gas.

■

cg̅ =

To solve this nonequilibrium BC mathematical model (eqs 5−8), the Laplace transform method is used. Using cg̅ to represent the Laplace transform of cg, the following expression is obtained:

∫0

y̅ =

.

f (s ) g (s )

(A.6)

where f(s) and g(s) are polynomials, and the degree of f(s) is lower than that of g(s). Then, the y in real space can be obtained as follows:53

−st

cg e dt

s Deff

(A.5)

If a transform y ̅ has the form of a quotient of polynomials, then the following can be expressed:

Derivation of the General Solution

cg̅ =

k intsqcosh(hq) + s 2 sinh(hq)

where q =

APPENDIX A

∞

cg,intk intqcosh(q(h − z))

(A.1)

The governing equation (eq 5) in the Laplace domain is as follows:

y(t ) =

sf (ar ) |p→0 + g (ar )

n

∑ r=1

f (ar ) ar t e |p≠0 g ′(ar )

(A.7)

2

Deff

d cg̅ = scg̅ dz 2

where ar is the rth positive zero root of polynomial g(s). As Carslaw54 did for the problem of heat conduction and Yang et al.45 did for the problem of modeling gas transport and storage in shale, eq A.5 is substituted into eq A.7 and the general analytical solution of the nonequilibrium BC model can be obtained:

(A.2)

The boundary conditions (eq 7 and eq 8) in the Laplace domain are given by

d cg̅ dz

|z = h , t ≥ 0 = 0

−Deff

d cg̅ dz

|z = 0, t ≥ 0

⎛ cg = cg,int⎜⎜1 − ⎝

(A.3)

⎛ cg,int ⎞ = k int⎜ − cg̅ |z = 0, t ≥ 0 ⎟ ⎝ s ⎠

(A.4)

∞

∑ n=1

where ξ is equal to

The solution for this ordinary differential equation in the Laplace domain can be obtained by substituting the boundary conditions of eq A.3 and eq A.4 into the governing equation (eq A.2) to yield the following expression:

2 2 2ξ cos(bn(h − z)/h)e−bn Deff t / h ⎞⎟ ⎟ (bn2 + ξ 2 + ξ)cos(bn) ⎠

hk int Deff

(A.8)

and bn is the nth positive nonzero root

of: bn tan(bn) = ξ I

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.7b01088. Digitalized experimental data using the direct methods (Table S1 and S2); four groups of experimental results using constant-pressure technique under 8.00 and 10.00 MPa (Figure S1) (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Mingzhe Dong. Tel: +1 403 210 7642; Fax: +1 403 284 4852. E-mail: [email protected]. ORCID

Mingzhe Dong: 0000-0002-2926-5139 Houjian Gong: 0000-0002-3304-1605 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support from the National Basic Research Program of China (No. 2014CB239103), the National Natural Science Foundation of China (No. 51204197 and 51274225), and Innovation Fund Designed for Graduate Students (YCX2017018) are gratefully acknowledged.

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K

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